(See attached file for full problem description)
Suppose that A, B, and AB are normal matrices. Prove that BA is also normal.
Here some hints: the trace of matrices can be used in clever ways to prove equalities.
Note that tr(A+B)=tr(A) + tr(B), tr(AB)=tr(BA) for any square A and B, and tr (C*C) 0 with equality if and only if C=0.There may be other techniques for doing this as well .
You can use to the theorem : Let A a square matrix then A is normal if only if there is a polynomial g such that A*=g(A). Where A* is the conjugate transpose.
I don't know what is better for you but when you prove this problem please be sure to use
the appropriate theorem or tool and explain with detail the prove, Thank you
This shows how to prove a matrix is normal.